Finite Volume Element Approximations Research Articles

A finite volume approximations for one nonlinear and nonlocal integrodifferential equations

In this paper, the error analysis of the Petrov–Galerkin finite volume element method (FVEM) is investigated for a nonlinear parabolic integro-differential equation that arises in the mathematical modeling of the penetration of a magnetic field into a substance, accounting for temperature-dependent changes in electrical conductivity. Starting from Maxwell’s equations, we derive a one-dimensional model problem, which forms the basis of our analysis. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. For simplicity, we consider only the lowest-order (linear and L-splines) finite volume elements. The novel contribution lies in the application of FVEM to this problem, leading to the establishment of an unconditionally stable numerical scheme and the derivation of optimal error estimates in the L∞(L2(Ω)) and L2(H01(Ω)) norms for both semi-discrete and linearized backward Euler fully-discrete schemes, using a generalized projection method that carefully manages the nonlinear terms. Lastly, numerical experiments are provided to support the theoretical conclusions.

Finite volume element approximation for time dependent convection diffusion reaction equations with memory

Error estimates for element schemes for time-dependent for convection-diffusion-reaction equations with memory are derived and stated. For the spatially discrete scheme, optimal order error estimates in L2 , H1 , and W1,p norms for 2 ≤ p < ∞, are obtained. In this paper, we also study the lumped mass modification. Based on the Crank-Nicolson method, a time discretization scheme is discussed and related error estimates are derived.

Error estimates of finite volume method for Stokes optimal control problem

In this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal L^{2}-norm error estimates. The approximate orders for the state, costate, and control variables are O(h^{2}) in the sense of L^{2}-norm. Furthermore, we derive H^{1}-norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.

Error estimates in $ L^2 $ and $ L^\infty $ norms of finite volume method for the bilinear elliptic optimal control problem

<abstract> This paper discusses some a priori error estimates of bilinear elliptic optimal control problems based on the finite volume element approximation. A case-based numerical example serves to discuss with optimal $ L^2 $-norm error estimates and $ L^{\infty} $-norm error estimates, and supports two key insights. First, the approximate orders for the state, costate and control variables are $ O(h^2) $ in the sense of $ L^{2} $-norm. Second, the approximate orders for the state, costate and control variables are $ O(h^2\sqrt{|lnh|}) $ in the sense of $ L^{\infty} $-norm. </abstract>

Two-Level Iterative Methods for Solving the Saddle Point Problems

Iterative processes in the Krylov subspaces for solving large ill conditioned saddle-type SLAEs with sparse matrices arising in finite difference, finite volume, and finite element approximations of multidimensional boundary value problems with complex geometric and functional properties of the initial data, characteristic of many relevant applications are studied. Combined two-level iterative algorithms using efficient Chebyshev acceleration and variational the conjugate directions methods, as well as the Golub-Kahan bi-diagonalization algorithms in the Krylov subspaces are considered. Examples of two-dimensional and three-dimensional filtration problems are used to study the resource consumption and computational performance of the proposed algorithms, as well as their scalable parallization on the multiprocessor systems with distributed and hierarchical shared memory.

Fourier Finite Volume Element Method for Two Classes of Optimal Control Problems Governed by Elliptic PDEs on Complex Connected Domain

In this article, Fourier finite volume element method is provided to solve two classes of optimal control problems governed by elliptic PDEs. In two classes of problems, both distributed control and Dirichlet boundary control are given. We obtain the optimality systems of optimal control problems by the Lagrange multiplier method. Using the polar coordinates, the Fourier discretization is used in the azimuthal direction while the finite volume element approximation is used in the radial direction. In our proposed numerical method, the trial and test function spaces are carefully chosen to get accurate approximation. We use the Fourier finite volume method to approximate the control problems, the two-dimensional optimal control problem can be reduced to a one-dimensional problem. We not only observe that the convergent order of distributed control is second order, but also the convergent order of the Dirichlet boundary control is second order. Some numerical experiments are presented to illustrate accuracy and efficiency of our method.

Finite Volume Element Approximation for the Elliptic Equation with Distributed Control

In this paper, we consider a priori error estimates for the finite volume element schemes of optimal control problems, which are governed by linear elliptic partial differential equation. The variational discretization approach is used to deal with the control. The error estimation shows that the combination of variational discretization and finite volume element formulation allows optimal convergence. Numerical results are provided to support our theoretical analysis.

Two-Level Least Squares Methods in Krylov Subspaces

Two-level least squares acceleration approaches are applied to the Chebyshev acceleration method and the restarted conjugate residual method in solving systems of linear algebraic equations with sparse unsymmetric coefficient matrices arising from finite volume or finite element approximations of boundary-value problems on irregular grids. Application of the proposed idea to other iterative restarted processes also is considered. The efficiency of the algorithms suggested is investigated numerically on a set of model Dirichlet problems for the convection-diffusion equation.

A stabilized finite volume method for the evolutionary Stokes–Darcy system

In this paper, we propose a stabilized fully discrete finite volume method based on two local Gauss integrals for a non-stationary Stokes–Darcy problem. This stabilized method is free of stabilized parameters and uses the lowest equal-order finite element triples P1–P1–P1 for approximating the velocity, pressure and hydraulic head of the Stokes–Darcy model. Under a modest time step restriction in relation to physical parameters, we give the stability analysis and the error estimates for the stabilized finite volume scheme by means of a relationship between finite volume and finite element approximations with the lower order elements. Finally, a series of numerical experiments are provided to demonstrate the validity of the theoretical results.

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Abstract The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg–de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg–de Vries equation.

A priori error estimates of finite volume method for nonlinear optimal control problem

In this paper, we study a priori error estimates for the finite volume element approximation of nonlinear optimal control problem. The schemes use discretizations based on a finite volume method. For the variational inequality, we use the method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate and control variables is O(h 2) or $$O\left( {{h^2}\sqrt {\left| {\ln h} \right|} } \right)$$ in the sense of L 2-norm or L ∞-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.

Superconvergence and asymptotic expansions for bilinear finite volume element approximation on non-uniform grids

We initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise for the isoparametric bilinear finite volume element scheme by employing the energy-embedded method on some non-uniform grids. Furthermore, we prove that the approximate derivatives are convergent of order two. Numerical examples confirm theoretical results.

A priori error estimates of finite volume method for nonlinear optimal control problem

In this paper, we study a priori error estimates for the finite volume element approximation of nonlinear elliptic and parabolic optimal control problem. The Schemes use discretizations base on a finite volume method. For the variational inequality, we use the method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate and control variables was obtained in the sense of some norms. Some numerical experiments are presented to test the theoretical result. Finally, we give some conclusions and future works.

Numerical method of mixed finite volume-modified upwind fractional step difference for three-dimensional semiconductor device transient behavior problems

Transient behavior of three-dimensional semiconductor device with heat conduction is described by a coupled mathematical system of four quasi-linear partial differential equations with initial-boundary value conditions. The electric potential is defined by an elliptic equation and it appears in the following three equations via the electric field intensity. The electron concentration and the hole concentration are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite volume element approximation, keeping physical conservation law, is used to get numerical values of the electric potential and the accuracy is improved one order. Two concentrations and the heat conduction are computed by a fractional step method combined with second-order upwind differences. This method can overcome numerical oscillation, dispersion and decreases computational complexity. Then a three-dimensional problem is solved by computing three successive one-dimensional problems where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by using prior estimate theory and other special techniques of partial differential equations. This type of mass-conservative parallel method is important and is most valuable in numerical analysis and application of semiconductor device.

A Priori Error Estimates of C rank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h2 + k2) is proved for the numerical solution of the control, state and adjoint-state in a discrete L2-norm. To derive this result, we also get the error estimate (convergent order O(h2 + k2)) of Crank–Nicolson finite volume element approximation for the second-order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016

Two-grid Methods for Finite Volume Element Approximations of Nonlinear Sobolev Equations

ABSTRACTIn this article, two-grid methods are studied for solving nonlinear Sobolev equation using the finite volume element method. The methods are based on one coarse grid space and one fine grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H), and the fine grid solution (with grid size h) can be obtained in a single symmetric and linear step. The optimal H1 error estimates are presented for the proposed methods, which show that the two-grid methods achieve optimal approximation as long as the mesh sizes satisfy h = 𝒪(H3|ln H|). As a result, solving such a large class of nonlinear Sobolev equations will not be much more difficult than solving one linearized equation.

Symmetric finite volume element approximations of second order linear hyperbolic integro-differential equations

In this paper, based on barycenter dual meshes, we develop one semi-discrete and two full discrete symmetric finite volume element schemes for second order linear hyperbolic integro-differential equations. The optimal order error estimates in L2 and H1-norms are derived for the semi-discrete scheme. Numerical experiments confirm the performance of the symmetric schemes, and further show that the L2-norm convergence rate of the full discrete backward Euler and Crank–Nicolson schemes to be O(h2+τ) and O(h2+τ2), respectively.

Finite volume element approximation of the coupled continuum pipe‐flow/Darcy model for flows in karst aquifers

A finite volume element method is applied to approximate the continuum pipe‐flow/Darcy problem, which models the coupled conduit flow and porous media flow in Karst aquifers. A decoupled scheme is proposed for solving the coupled discretization problem. Optimal error estimates in L2 norm and H1 norm are given in this article. Some numerical examples are presented to verify the theoretical results and demonstrate the effectiveness of the decoupling approach. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 376–392, 2014

Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion

This paper is concerned with the study of pattern formation for an inhomogeneous Brusselator model with cross-diffusion, modeling an autocatalytic chemical reaction taking place in a three-dimensional domain. For the spatial discretization of the problem we develop a novel finite volume element (FVE) method associated to a piecewise linear finite element approximation of the cross-diffusion system. We study the main properties of the unique equilibrium of the related dynamical system. A rigorous linear stability analysis around the spatially homogeneous steady state is provided and we address in detail the formation of Turing patterns driven by the cross-diffusion effect. In addition we focus on the spatial accuracy of the FVE method, and a series of numerical simulations confirm the expected behavior of the solutions. In particular we show that, depending on the spatial dimension, the magnitude of the cross-diffusion influences the selection of spatial patterns.

Superconvergence and Asymptotic Expansions for Bilinear Finite Volume Element Approximations

Superconvergence and Asymptotic Expansions for Bilinear Finite Volume Element Approximations